Fluids such as human blood demonstrate several non-Newtonian properties that can render classic rheological models insufficient for effective analysis. The unique nature of blood’s viscosity and elasticity is in large part due to the fluid’s suspended microstructures such as red blood cells, white blood cells, and platelets. These collectively lead blood to demonstrate the shear-thinning, viscoelastic, yield stress, and thixotropic behaviors characteristic of complex materials. Importantly, red blood cells (RBC) present the predominant portion of the microstructures suspended in the blood plasma and are thus of utmost significance in affecting blood’s fluid properties. RBCs have been shown to agglomerate into coin-stack like structures through fibrinogen fusing at low shear rates1,2. In observing this shear-dependent formation and breakdown phenomena of these rouleaux, the thixotropic nature of blood becomes further evident. In such fluid, complex materials, microstructure such as rouleaux can interact with the solvent medium to dramatically alter the behavior of the material.
The status of blood as a thixo-elasto-visco-plastic (TEVP) fluid entails notable differences in the methods by which its peculiar rheological behavior might be analyzed3. A notable feature of some TEVP materials is shear thinning, describing a tendency for fluid viscosity to decrease under mounting shear stresses4,5. This phenomenon can be observed in the disintegration of blood’s rouleaux microstructures, lending the fluid a more free manner of flow6. A comprehensive understanding of shear thinning is necessary to better characterize its behavior in natural, necessary behaviors such as blood clotting, where blood viscosity increases dramatically near wounds to seal breaches in the circulatory system. In addition to body health and blood composition, certain pharmaceuticals have been observed to affect blood shear properties. The evolution of this shear thinning can potentially induce stress in the vascocirculatory system as the body attempts to maintain a state of equilibrated flow throughout1,3,5,7,8.
As a TEVP material, blood also possesses a yield stress that ensures that its cells do not deform with some degree of stress application, were Newtonian fluids would, conversely, be inclined to deform without consistent application of a preserving force. This complex behavior entails that the cardiovascular system must continually apply work to maintain the blood yield necessary for healthy flow6. Relevant to this effort, ex-vivo experimentation on work-induced fluid deformation properties can be prosecuted via the use of a rheometer7. In addition to blood’s inherent material properties, the present microstructure can contribute to total yield stress characteristics. This relates to the thixotropic aspect of TEVP materials such as blood, which describes the evolution and devolution of the rouleaux microstructure at low and high shears respectively7,9. The elastic component refers to blood’s pre-deformation elastic properties8,10,11. The last component, plasticity, hails to the plastic nature of the rouleaux, which can deform and given a certain degree of applied shear, undergo irreversible change4,5.
In the past years, classical steady-state modeling has undergone several evolutions, with the Casson, Carreau-Yasuda, Bingham, and Herschel-Bulkley models proving particularly notable frameworks. However, these simple models lack the ability to deliberately and accurately characterize the evolving nature of the steady-state and transient flow regimes that are present in TEVP materials such as blood. The thixotropic aspect of complex materials generally acts to reduce the adherence of the model to collected data, especially at lower strain amplitudes. This fact resulted in the development of a new generation of models integrating systems of timescale-based differential equations and constraints to better describe the nuances of TEVP fluids1,3,4,8,12,13,14,15,16,17,18,19,20,21,22,23.
While initial enhanced models were simply classical models outfitted with additional functions and parameters, models such as that proposed by Dullaert and Mewis completely novel methods by which to analysis complex materials, grounded within the peculiar physical processes that distinguish TEVP fluids. Shared by these models and other Maxwellian are three primary thixotropic features: shear breakage representing the dissolution of microstructure like blood rouleaux, shear aggregation describing the inter-microparticle interactions of a fluid at a certain shear rate, and Brownian aggregation. The Brownian aggregation term aims to somewhat characterize the way RBCs spontaneously aggregate due to the random nature of Brownian motion3,4,12.
The thixotropic components of these novel models generally utilized three modes of structure dynamics: shear structure breakage, structure reconstitution due to Brownian forces, and shear aggregation1,3,4,8,14,15,16,17,18,19,20,21,22,23. Some newer thixotropic models, such as that developed by Wei and Solomon, incorporated these structure dynamics, incorporating a function for back-stress derived from the material microstructure24,25. In more accurately representing the formation and dissolution of microstructure with varying shear rate, such models can more accurately model microstructure under steady-state and transient flow regimes. In these models, the structure parameter, λ, describes a fully structured material at a value of 1 where each particle enjoys a full range of connections with its neighbors. However, as λ approaches 0, the microstructure decays14,15,21. This parameter allows for a more comprehensive representation of the microstructure deformation that can occur in complex materials with changing shear rate. This deformation leads to viscoelastic activity within the material, entailing a measurable stress response3,8,15,21,22,26. The model’s elasticity calculation also changes with this evolution in structure, necessitating correction of the model’s yield stress and elastic modulus16,17,26. As such, the total stress is the sum of that of the solvent structure and microstructure. Additionally, the model defines two types of viscosity in a flow state: viscosity due to variable microstructure ληST and pure solvent viscosity η∞14,15. By accounting for the various elements of the total viscosity, the model can more effectively represent viscosity evolution under transient shear rate conditions14,15.
Furthermore, blood’s viscoelasticity necessitates the depiction of microstructure-dependent elasticity via dual elastic and plastic stress components or, alternatively, the viscoelastic model’s inclusion of the structure parameter17,18. The former method involves the separation of the total strain and its time derivative into two, independent functions12,17,21. The kinematic hardening theories of plasticity, encompassing isotropic hardening (IH) and kinematic hardening (KH) can be applied to model plastic behavior, can be applied to modeling a material’s plastic behavior. The IH is relevant not just to plasticity but also the material’s thixotropic properties, as described by a dimensionless, internal structure parameter. KH describes the effective yield stress as a function of deformation and can induce delays between back stress evolution and shear stress5,12,41. Inclusion of IH and KH in modeling TEVP systems is vital though not necessarily sufficient for a comprehensive model of viscoelastic behavior, necessitating the addition of thixotropic structure parameters, kinetic equations, and viscoelasticity.
The separation of thixotropic response, viscoelastic response, shear structure breakage and structure build-up into separate timescales in models such as ethixo-mHAWB best allows the accurate replication of blood’s rheological behavior and the rouleaux within3,44. While this does add more parameters, contributing to model complexity, the new thixotropic and viscoelastic timescales provide vital insight into the evolution of the bloodstream’s rouleaux. CFD modeling can then be used in conjunction with such enhanced models for more effective blood analysis.
While a basic depiction of kinematic hardening had been present in earlier thixotropic models, the ability to test the comprehensiveness of newer TEVP models was hamstrung by a relative lack of detailed transient experimental data. However, as more rigorously collected data, has become available in recent years, progress on the further development of thixotropic models has become possible1,8. This new data extended beyond that of the steady-state, including variable amplitude and frequency datapoints from UD-LAOS analysis, drawn from the superposition of steady and oscillatory shear of the thixotropic model. This best allows for the representation of the RBCs’ viscoelastic feature via the use of a generalized White–Metzner-Cross model, producing a thixotropic viscoelastic model (TVM or, formerly, HAWB)8. The improved mHAWB variant of the model would manifest through the incorporation of a rouleaux viscoelastic response into TVM3. The mHAWB model itself has seen several modifications and improvements, with new elements integrated to produce the ETV and ESSTV models3,44.
Recent research has extended to the exploration of different models, with Armstrong and Tussing and Armstrong and Pincot investigating the potential use of the Oldroyd-8 and Giesekus models, respectively, to describe RBC behavior as an alternative to the ubiquitous generalized White–Metzner-Cross framework as used in the mHAWB model35,36,37,38.However, other research by Armstrong et al. delved into the potential use of Saramito’s Herschel-Bulkley model to better portray blood rheology while discounting the viscoelastic nature of deformable RBCs to produce the ethixo elastoviscoplastic (ethixo EVP) model. The ethixo EVP model integrated thixotropy trough the inclusion of a structural parameter bound to a kinetic equation to better represent the complexity of TEVP fluid39. The addition of thixotropy was shown to better the fit the model to the transient experimental blood rheological data.
A different approach taken in adding viscoelastic features to established viscoplastic and thixotropic models also entailed a better fit to the collected data1,5. As developed by Wei et al., the ML-IKH model provided an array of lambda values, featuring independent thixotropic evolution timescales, parallel to an isotropic kinematic hardening framework40. An analogous effort was accomplished in the modification of the SPTT-Isotropic Kinematic Hardening model (“S” for Saramito’s novel plasticity term, “PTT” for the Phain-Thien Tanner viscoelastic model) which acted to combine several approaches to effectively representing the material physics of a TEVP materal17,41. These models possessed a tensorial form and possessed 11 to 15 distinct parameters. Concurrent to the developments, the Modified Delaware Thixotropic Model (MDTM) was enhanced with a viscoelastic timescale of the stress response contribution from the component rouleaux, the novel model being dubbed viscoelastic enhanced MDTM (VE-MDTM)23,31,35,42,43. This alteration further demonstrated the importance of including the stress response from changes in microstructure. Recent work has also extended toward the tensorial transformation ETV and ESSTV model into tensorial analogues: t-ETV and t-ESSTV respectively45.
Despite the relative effectiveness of current generation models in fitting steady state and, most notably, transient rheological data for blood, their relative efficiency in analyzing the material nature of blood does leave something to be desired5,10. This effort utilizes elements from the previously established steady-state variants of the Dullaert & Mewis and MDTM models. These are then recast into a dynamic Maxwellian format, allowing for a tensorial representation17,41. The development of this new system, dubbed the tensorial-enhanced-Thixo-Visco-Plastic (t-e-TVP) model, is fully described in “Model development” section, where the new method is shown to integrate theories of plasticity to better express the elastic and viscoelastic contributions of the blood rouleaux towards total stress and integrate the full stress tensor17,40,41. “Methods” section follows with a description of the experimental protocol relevant to the collection of the experimental samples and a walkthrough of the parametric optimization performed to fit the experimental model to the given data. “Results and discussion” section includes a summary of the experimental results, analyzing the capabilities of the model to predict large amplitude oscillatory shear and uni-directional large amplitude oscillatory shear flow in the circulatory system. The accuracy of the novel t-e-EVP framework in predicting SAOS, LAOS, and UD-LAOS is then compared to that the t-ethixo-mHAWB variant, representing one of the most modern contemporary enhanced rheological modes. The conclusions of the analysis are then enumerated in “Conclusions” section.
